Black Hole

Prerequisites: General Relativity. Full path (16 pages)

Best explanations

Intuitive How We Know Black Holes Exist

Intuitive
Concrete
Abstract
Why is it interesting?
History

Intuitive

A black hole is a region of space and time where light cannot escape because gravity is so strong. This is possible because light always moves with the same velocity in the vacuum and therefore cannot move so fast as would be required to escape the black hole. In addition, no object or thing can move faster than with this speed of light, and therefore nothing can escape a black hole.

As a result, no light and nothing else can emerge from the black hole.

However, light and other objects can fall into the black hole. If this happens, the black hole gets heavier and larger.

The surface of a black hole is called the black hole horizon.

A black hole is created when a very massive object, like a huge star, collapses. This happens, for example, when a star is burned out and no longer has enough "nuclear fuel". As long as the star is able to radiate, this radiation works against the gravitational pull and keeps the star stable. But when it no longer can radiate, the whole mass of the star collapses due to gravity. The original mass of the object will compress infinitely and forms an extremely massive object that we call a black hole.

It is important to note that the horizon is not the surface that formed the black hole. Instead, the horizon is the boundary of the region where no light can escape. Gravity gets weak as we move away from a massive object. Therefore, the farther we are from a massive object, the weaker gravity gets. As a result, if we move far enough away from a black hole its gravitational pull gets so weak that light can escape again. At the horizon is the surface where gravity is only just strong enough to hold everything inside. Outside of the horizon, light can escape.

A great discussion of Black Holes in layperson terms can be found in Chapter 5 of Three Roads to Quantum Gravity by Lee Smolin
See also the short video series by Minutephysics:
See also Everything You Ever Wanted to Know About Black Holes by Dank Falk

Concrete

Black Hole Entropy - Bekenstein's law

Hawking discovered that the area of the horizon of a black hole can never decrease in time. In this sense, it is analogous to entropy, which also always gets larger and larger.

The great insight of Bekenstein was that this was not just an analogy. He argued that a black hole has real entropy, which he conjectured is proportional to the area of its horizon and measures the amount of information trapped beyond that horizon. Three Roads to Quantum Gravity by Lee Smolin

The entropy $S$ of a black hole is $S= \frac{A}{4 G \hbar}$ (in units with $c=k_B=1$ ), where $A$ is the area of the black hole horizon and $G$ the gravitational constant.

here is a very simple way to interpret this equation which is due to Gerard 't Hooft, who did important work in element- ary particle physics ± for which he won the 1999 Nobel Prize for Physics ± before turning his attention to the problem of quantum gravity. He suggests that the horizon of a black hole is like a computer screen, with one pixel for every four Planck areas. Each pixel can be on or off, which means that it codes one bit of information. The total number of bits of information contained within a black hole is then equal to the total number of such pixels that it would take to cover the horizon. The Planck units are very, very tiny. It would take 10 66 Planck area pixels to cover a single square centimetre. So an astrophysical black hole whose horizon has a diameter of several kilometres can contain a stupendous amount of information. Three Roads to Quantum Gravity by Lee Smolin

Black Hole Temperature - Hawking Radiation

see also Hawking Radiation

In the beginning, many people did not take Bekenstein's ideas about the entropy of a black hole seriously, because of the following reason:

If nothing can come out of a black hole, then it has zero temperature. This is because temperature measures the energy in random motion, and if there is nothing in a box, there can be no motion of any kind, random or otherwise. But ordinarily a system cannot have entropy without it being hot. This is because the missing information results in random motion, which means there is heat. So, if black holes have entropy, there is a violation of the laws of thermodynamics. Three Roads to Quantum Gravity by Lee Smolin

However:

The mystery was solved first by Hawking, who realized that if black holes were hot there was no contradiction with the laws of thermodynamics. By following a chain of reasoning roughly like the story above, he was able to show that an observer outside a black hole would see it to be at a finite temperature. Expressed in Planck units, the temperature $T$ of a black hole is inversely proportional to its mass, $m$ . Three Roads to Quantum Gravity by Lee Smolin

$T= \frac{\hbar c^3}{8 \pi G M k_B} ,$

where $k_B$ is the Boltzmann constant, $c$ the speed of light, $G$ the gravitational constant, $\hbar$ the reduced Planck constant and $M$ the mass of the black hole.

This formula shows why black holes are so important and interesting. In this little formula everything comes together:

Quantum mechanics, in the form of $\hbar$
Gravity, in the form of $G$

In this sense, black holes are laboratories for quantum gravity.

The temperature of a black hole is tiny. Putting in the numbers yields

$T= 6.169 \cdot 10^{-8} \text{ K } \ \frac{M_\odot }{M},$ where $M_\odot$ is the mass of the sun. In words this means that black hole with a mass equal to the mass of our sun would have a temperature of only $10^{-8}$ K. If the black hole is heavier, the temperature gets even tinier.

Far from a black hole, spacetime is "flat" and you can calculate the configuration of quantum fields that we think of as "the vacuum" (empty space).

You can do the same thing for "empty space" near the edge of a black hole.

However, what Hawking did was show that the "empty space/vacuum" configuration near the event horizon was *different* from the one far from the black hole. Moreover if you started with fields in the vacuum configuration near the black hole, far from the black hole, they'd look exactly like a thermally radiating system. That is, by bending spacetime, the black hole made the vacuum itself radiate away energy. The source of this energy is the black hole's mass, and eventually this process (Hawking radiation) would cause the black hole to evaporate. And there is still so much we don't understand about what all this means. We still don't really know where the entropy of a black hole is coming from. But Hawking made the connection explicit: there is a black hole entropy, it's proportional to the surface area, and it's enormous. […] (There's another way to describe Hawking radiation as "virtual particles near the horizon, one falls in, the other escapes." That's not even really morally incorrect, but its slightly more accurate to think of the vacuum changing.) Anyway, Hawking radiation allows one to show that the maximum entropy ANY volume can have is the entropy of a black hole contained in the volume. Since the entropy of a black hole is set by the surface area, this means the maximum entropy of a volume goes like the area. That's sort of nuts, because it means if I make a room 2x larger in each dimension, I can fit $2^3=8$ times as much stuff, but only 2^2=4 times as much entropy ("information" loosely speaking). So somehow, the information in volume is encoded in the surface area of the volume. This is the "holographic principle" and I don't think we have a great idea of what it means about how our Universe is put together. Anyway, Hawking radiation is a fascinating thing. And it is the opening step in a really interesting area of physics that seems to be saying something very deep about how the Universe works. I'm not sure what.

Matthew Buckley

Abstract

We use units $G = c = 1$ in this section, and the signature $(-,+,+,+)$ .

Definition. Let $(M, g)$ be a time-orientable Lorentzian manifold that is asymptotically flat, with future null infinity $\mathscr I^+$ in its conformal compactification. The black hole region is the set of points from which no future-directed causal curve reaches $\mathscr I^+$ : $\mathcal B = M \setminus J^-(\mathscr I^+),$ where $J^-$ denotes the causal past. The event horizon is its boundary $\mathcal H = \partial J^-(\mathscr I^+)$ . It is a null hypersurface (achronal and generated by null geodesics wherever it is smooth). The definition is global: whether a point lies inside the black hole depends on the entire future of the spacetime, so the horizon cannot be located by local measurements.

Examples. The Schwarzschild solution $ds^2 = -\left(1 - \frac{2M}{r}\right)dt^2 + \left(1 - \frac{2M}{r}\right)^{-1} dr^2 + r^2 d\Omega^2$ has its horizon at $r = 2M$ . The apparent singularity there is a coordinate artefact, removed for example by Kruskal coordinates, while $r = 0$ is a curvature singularity where the Kretschmann scalar $R_{abcd}R^{abcd} = 48M^2/r^6$ diverges. The uniqueness theorems state that, under assumptions of stationarity, asymptotic flatness, regularity of the domain of outer communication and Einstein-Maxwell field equations, the black hole is described by the Kerr-Newman family with parameters mass $M$ , angular momentum $J$ and charge $Q$ .

Singularities. Penrose's singularity theorem states that if a spacetime with a non-compact Cauchy surface contains a closed trapped surface and satisfies the null energy condition $R_{ab}k^a k^b \ge 0$ for null $k^a$ , then it is future null geodesically incomplete. So once gravitational collapse has formed a trapped surface, a singularity in the sense of geodesic incompleteness cannot be avoided, without any assumption of symmetry. That the singularity is hidden behind a horizon is the content of the weak cosmic censorship conjecture, which is unproven.

Black hole mechanics. For a stationary black hole the horizon is a Killing horizon of a Killing field $\xi^a = t^a + \Omega_H \phi^a$ , and the surface gravity $\kappa$ is defined by $\xi^b\nabla_b \xi^a = \kappa\, \xi^a$ on $\mathcal H$ . The laws of black hole mechanics read:

Zeroth law: $\kappa$ is constant over the horizon of a stationary black hole.
First law: $\delta M = \frac{\kappa}{8\pi}\,\delta A + \Omega_H\, \delta J + \Phi_H\, \delta Q$ .
Second law: if the null energy condition holds (and the spacetime is suitably regular), the horizon area $A$ never decreases.

With $\hbar$ restored and quantum fields on the background, Hawking's result identifies the temperature $T = \hbar\kappa/(2\pi k_B)$ and hence, through the first law, the entropy $S = k_B A/(4\hbar)$ (in units $G = c = 1$ ). For Schwarzschild, $\kappa = 1/(4M)$ , which reproduces $T = \hbar c^3/(8\pi G M k_B)$ given above.

Why is it interesting?

Among those of us who think about space and time professionally, black holes play a central role. […] Beyond the fascination they hold for astronomers, black holes are important to science for other reasons. They are a central object of study for those of us who work on quantum gravity. In a sense, black holes are microscopes of infinite power which make it possible for us to see the physics that operates on the Planck scale.

Three Roads to Quantum Gravity by Lee Smolin

History

Jacob Bekenstein and the Development of Black Hole Thermodynamics by Robert M. Wald

Path to this page

Leads to: Hawking Radiation.